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segmented circle

The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane along x & y axes. Thus four identical segments of the circular plane are left over.

Question: How to calculate the solid angle subtended by the remaining plane (consisting of four identical segments) at a point $P (0, 0, 4cm)$ lying on the z-axis (normally outwards to the plane of paper)?

Solid angle $(\omega)$ subtended by the circular plane, with a radius $r$ at any point lying on the geometrical axis at a distance $h$ from the center, is given as $$\omega=2\pi \left(1-\frac{h}{\sqrt{h^2+r^2}}\right)$$ And the solid angle $(\omega)$ subtended by any rectangle of size $l*b$ at any point lying on the perpendicular axis at a distance $d$ from the center is given by standard formula as $$\omega=4sin^{-1}\left(\frac{lb}{\sqrt{(l^2+4d^2)(b^2+4d^2)}}\right)$$

But how to apply these formula on the non radial segments of a circle? Any help is greatly appreciated.

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The distance d=4 cm, radius R=7cm and gap g=2cm give a solid angle of 1.70538588 if we use my formula (C6) in Solid angle of a Rectangular Plate.

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    $\begingroup$ The solution seems to be very lengthy, don't you have any shortcut method to deal with such problems? $\endgroup$ May 17 '15 at 19:28
  • $\begingroup$ Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$
    – user642796
    May 18 '15 at 7:58

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